In this paper we analyze the approximation, by piecewise linear finite elements, of a Steklov eigenvalue problem in a plane domain with an external cusp. This problem is not covered by the literature and its analysis requires a special treatment. Indeed, we develop new trace theorems and we also obtain regularity results for the source counterpart. Moreover, under appropriate assumptions on the meshes, we present interpolation error estimates for functions in fractional Sobolev spaces. These estimates allow us to obtain appropriate convergence results of the source counterpart which, in the context of the theory of compact operator, are a fundamental tool in order to prove the convergence of the eigenpairs. At the end, we prove the convergence of the eigenpairs by using graded meshes and present some numerical tests.

中文翻译：

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尖角域中的 Steklov 特征值问题

在本文中，我们通过分段线性有限元分析具有外部尖点的平面域中的 Steklov 特征值问题的近似。这个问题没有包含在文献中，它的分析需要特殊处理。事实上，我们开发了新的迹定理，并且我们还获得了源对应物的正则性结果。此外，在适当的网格假设下，我们提出了分数 Sobolev 空间中函数的插值误差估计。这些估计使我们能够获得源对应物的适当收敛结果，在紧算子理论的背景下，这是证明特征对收敛的基本工具。最后，我们使用分级网格证明了特征对的收敛性，并给出了一些数值测试。